Is 6^17 + 17^6 Divisible by 3 or 7?

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Homework Help Overview

The discussion revolves around the divisibility of the expression 6^17 + 17^6 by the numbers 3 and 7. Participants are exploring the properties of these numbers and their powers in relation to modular arithmetic.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • One participant attempts to demonstrate that 6^17 is divisible by 3 using modular arithmetic, while another questions the validity of the initial calculations and notation. There is mention of Fermat's little theorem as a potential approach, and a discussion about the implications of 17 being a prime number in the context of divisibility by 3.

Discussion Status

The discussion is ongoing, with participants raising questions about the calculations and assumptions made regarding the powers of 6 and 17. Some guidance has been offered regarding the use of modular arithmetic, but no consensus has been reached on the overall divisibility of the expression.

Contextual Notes

Participants are navigating through potential errors in notation and assumptions, particularly regarding the powers involved and their implications for divisibility. There is an emphasis on the properties of prime numbers in the context of the problem.

lordy12
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1. 6^17 + 17^6 is divisble by 3 or 7?

Homework Equations


3. 6^1 = 0(mod 3)
(6^1)^17=0(mod 3) so 6^17 is divisible by 3
how do u do this?
 
Last edited:
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17 to any power is prime... No.

What is the stray T in your first line. Where did 61 come from? Why are you raising it to the power 1?
 
sorry i changed it
 
Fermat's little theorem.
 
lordy12 said:
1. 6^17 + 17^6 is divisble by 3 or 7?



Homework Equations





3. 6^1 = 0(mod 3)
(6^1)^17=0(mod 3) so 6^17 is divisible by 3
how do u do this?

For 3 at least, you don't need anyone as powerful as Fermat! Suppose 617+ 176 were divisible by 3. Then we would have 617+ 176= 3n for some integer n. Then 176= 3n- 617= 3n- 317217= 3(n- 316217) which is impossible: since 17 is prime, no power of 17 is divisible by 3.
 

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